GAP/QPA algorithm to trivial extension

Question

I am studying trivial extensions of finite dimensional $K$-algebras. My reading is the work of Fernández and Platzeck, for example https://www.sciencedirect.com/science/article/pii/S0021869301990568.

The GAP/QPA is great to make some examples. I am learning much here, special thanks to Oeyvind Solberg. When I created some presentations using Fernández and Platzeck results, I didn’t get the same results as the QPA.

For example: Let $A = KQ_A / I_A$ be a fixed presentation for $K$-algebra $A$. We fix a set $\mathbb{M} = \{ p_1, p_2, \dotsc, p_t \}$ of elements in $KQ_A$ such that $\{ \bar{p_1}, \dotsc, \bar{p_t} \}$ is a basis for $soc_{A^e} A$, and for each $i$, $\beta_{p_i}$ is an arrow from $e(p_i)$ to $s(p_i)$.

Using Fernández and Platzeck results, $T(A)$ has a presentation

with the admissible ideal generated by:

• $a1 \beta_{p_1} a1$, $\beta_{p_1} a1 \beta_{p_1}$, $a2 a5 \beta_{p_2} a2$, $\dotsc$, $\beta_{p_2} a3 a4 \beta_{p_2}$;
• $a1 a2$, $a1 a3$, $\beta_{p_2} \beta_{p_1}$, $a5 \beta_{p_2} a3$ e $a4 \beta_{p_2} a2$;
• $a2 a5 - a3 a4$.

Using GAP/QPA, $T(A)$ has an isomorphic presentation:

gap> TA := TrivialExtensionOfQuiverAlgebra(A);  
<Rationals[<quiver with 5 vertices and 7 arrows>]/
<two-sided ideal in <Rationals[<quiver with 5 vertices and 7 arrows>]>, (10 generators)>>
gap> RelationsOfAlgebra(TA);  
[ (1)*a1*a2, (1)*a1*a3, (-1)*a2*a5+(1)*a3*a4, (1)*te_a1_2_2*te_a1_1_1, (1)*a1*te_a1_1_1*a1,
  (-1)*te_a1_1_1*a1+(1)*a2*a5*te_a1_2_2, (1)*a4*te_a1_2_2*a2, (1)*a5*te_a1_2_2*a3, (1)*te_a1_1_1*a1*te_a1_1_1,
  (1)*a5*te_a1_2_2*a2*a5 ]

where te_a1_1_1$=\beta_{p_1}$ and te_a1_2_2$=\beta_{p_1}$

My question is which algorithm the QPA uses to calculate the presentation of trivial extensions of finite dimensional algebras.

Again, I appreciate it.

Answer 1

I am sorry that I misunderstood your question. I thought that the theory and QPA gave two non-isomorphic answers.

Algorithm:

1. The quiver $Q_{T(A)}$ of $T(A)$ contains a copy of the original quiver, label the vertices and the arrows as in the original quiver.
2. Find the new arrows by finding the top of $D(A)$ as a bimodule.
3. Add the relations from the original algebra to new_relations.
4. Add te_arrows * te_arrows and te_arrows * paths in original arrows * te_arrows to new_relations.
5. Compute $B = kQ_{T(A)}/\langle \textrm{new_relations}, J^t\rangle$, for $t = \operatorname{LL}(A) + 2$.
6. "Find" $D(A) \hookrightarrow B \to D(A)\hookrightarrow T(A)$, and compute the kernel. This gives the relations extra_te_relationsinvolving the te_arrows.
7. Compute $kQ_{T(A)}/\langle \textrm{new_relation}, \textrm{extra_te_relations}\rangle$, which is the desired representation of $T(A)$ as a quotient of a path algebra.

For further details look at the very end of the file pathalgtensor.gi, in the lib directory of your GAP-installation (in the directory: gap-4.??.?/pkg/qpa/lib/).